Exponential ergodicity of the solutions to SDE's with a jump noise

0.38 MB·English

by  Alexey M. Kulik

EXPONENTIAL ERGODICITY OF THE SOLUTIONS TO SDE’S WITH A JUMP
NOISE
ALEXEYM.KULIK
Abstract. The mild sufficient conditions for exponential ergodicity of a Markov process, defined as the
solution to SDE with a jump noise, are given. These conditions include three principal claims: recurrence
7 condition R, topological irreducibilitycondition Sandnon-degeneracy condition N, the latter formulatedin
0 thetermsofacertainrandom subspaceofRm,associated withtheinitialequation. Theexamples aregiven,
0 showingthat,ingeneral,noneofthreeprincipalclaimscanberemovedwithoutlosingergodicityoftheprocess.
2 Thekeypointintheapproach,developedinthepaper,isthatthelocalDoeblincondition canbederivedfrom
n NandSviathestratificationmethodandcriteriumfortheconvergenceinvariationsofthefamilyofinduced
a measuresonRm.
J

7
2 Introduction

R] In this paper, we study ergodic properties of a Markov process X in Rm, given by an SDE
P
. (0.1) dX(t)=a(X(t))dt+ c(X(t−),u)ν˜(dt,du)+ c(X(t−),u)ν(dt,du).
h
Zkuk≤1 Zkuk>1
t
a
Here,ν isaPoissonpointmeasure,ν˜iscorrespondentcompensatedmeasure,andcoefficientsa,caresupposed
m
to satisfy usual conditions sufficient for the strong solution of (0.1) to exist and be unique. Our aim is to give
[
sufficient conditions for exponential ergodicity of (0.1), that impose as weak restrictions on the L´evy measure

2
of the noise, as it is possible.
v
There exists two well developed methods to treat the ergodicity problem for the discrete time Markov
7
4 processes, valued in a locally compact phase space. The first one is based on the coupling technique (see
7
detailedoverviewin[14]),thesecondoneusesthenotionsofT-chain andpetitesets (see[21],[6]andreferences
1
therein). These methods can be naturally extended to continuous time case either by making a procedure of
0
7 time discretization (like it was made for the solutions to SDE’s with jumps in the recent paper [20]), or by
0 straightforward use of the coupling technique in a continuous time settings (see [27],[28] for such kind of a
/
h technique for diffusion processes). Typically, in the methods mentioned above, the two principal features
t
a should be provided:
m – recurrence outside some large ball;
: – regularity of the transition probability in some bounded domain.
v
i The first feature can be provided in a quite standard way via an appropriate version of the Lyapunov
X
criterium (condition R in Theorem 1.1 below). The second one is more intrinsic, and requires some accuracy
r
in the choice both of the concrete terms, in which such feature is formulated, and of the conditions on the
a
process,sufficientforsuchfeaturetoholdtrue. We dealwiththeformoftheregularityfeature,thatisusually
called the local Doeblin condition, and is formulated below.
LD. For every R>0, there exists T =T(R)>0 such that
inf [PT ∧PT](dz)>0,
x y
kxk,kyk≤RZRm
2000 Mathematics Subject Classification. Primary60J25;Secondary 60H07.
Key words and phrases. β-mixingcoefficients, localDoeblincondition, admissibletime-stretchingtransformations, stratifica-
tionmethod,convergence invariationofinducedmeasures.
1
2 ALEXEYM.KULIK
where Pt(·)≡P(X(t)∈·|X(0)=x), and, for any two probability measures µ,κ,
x
df dµ dκ
[µ∧κ](dz)=min (z), (z) (µ+κ)(dz).
d(µ+κ) d(µ+κ)
(cid:20) (cid:21)
The non-trivialquestion is what is the proper form of the conditions on the coefficients a,c of the equation
(0.1) and the L´evy measure of the noise, sufficient for the local Doeblin condition to hold true. In a diffusion
settings, standard strong ellipticity (or, more general, Ho¨rmander type) non-degeneracy conditions on the
coefficients provide that the transition probability of the process possesses smooth density w.r.t. Lebesgue
measure, and thus LD holds true ([27],[28]). In a jump noise case, one can proceed analogously and claim
the transition probability of the solution to (0.1) to possess a locally bounded density (exactly this claim was
used as a basic assumption in the recent paper [20]). However,in the latter case such kind of a claim appears
to be too restrictive; let us discuss this question in more details. Consider, for simplicity, equation (0.1) with
c(x,u)=u, i.e. a following non-linear analogue of the Ornstein-Uhlenbeck equation:
(0.2) dX(t)=a(X(t))dt+dU ,
t
t t
where U = c(u)uν˜(ds,du)+ c(u)ν(ds,du) is a L´evy process. There exist two methods to
t 0 kuk≤1 0 kuk>1
provide the process defined by (0.2) to possess a bounded (moreover, belonging to the class C∞) transition
R R R R
probability density. The first one was initially proposed by J.Bismut (see [3],[2],[19],[15]), the second one –
by J.Picard (see [24],[13]). Both these methods require, among others, the following condition on the L´evy
measure Π of the process U to hold true:
(0.3) ∃ρ∈(0,2): ε−ρ kuk2Π(du)→∞, ε→0+.
Zkuk≤ε
This limitation is not a formal one. It is known (see [18], Theorem 1.4), that if
−1
1
(0.4) liminf ε2ln sup [|(u,v)|∧ε]2Π(du)=0,
ε→0+ (cid:20) (cid:18)ε(cid:19)(cid:21) kvk=1ZRm
then the transitionprobabilitydensity, if exists,does notbelong to anyL (Rm),p>1, andthereforeis not
p,loc
locally bounded (note that (0.4) implies that (0.3) fails). One can say that when the intensity of the jump
noise is ”sparse near zero” in a sense of (0.4), the behavior of the density essentially differs from the diffusion
one, and the density either does not exist or is essentially irregular.
On the other hand, let us formulate a corollary of the general ergodicity result, given in Theorems 1.1,1.3
below.
Proposition 0.1. Let m = 1, suppose that a(·) is locally Lipschitz on R and lim sup a(x) < 0. Suppose
x
|x|→+∞
that the L´evy measure Π of the process U satisfies the following conditions:
(i) there exists q >0: |u|qΠ(du)<+∞;
|u|>1
(ii) Π(R\{0})6=0.
R
Then the solution to (0.1) is exponentially ergodic, i.e. its invariant distribution µ exists and is unique,
inv
and, for some positive constant C,
(0.5) ∀x∈R, kPt−µ k =O(exp[−Ct]), t→+∞.
x inv var
In this statement, the non-degeneracy condition (ii) on the jump noise, obviously, is the weakest possible
one: if it fails, then (0.2) is an ODE, and (0.5) fails also. We can conclude, that the proper conditions on the
jump noise, sufficient to provide exponential ergodicity of the process, defined by (0.2), are much milder than
the conditions thatshouldbe imposedinorderto providethatthis processpossessesregular(locally bounded
or even locally L -integrable) transition probability density.
p
OurwaytoprovethelocalDoeblinconditionforthesolutionto(0.1)stronglyreliesonthefinite-dimensional
criterium for the convergence in variations of the family of induced measures on Rm. This criterium was
obtainedin[1](thecasem=1wastreatedin[7]). Viathestratification method (forthedetailedexpositionof
EXPONENTIAL ERGODICITY OF THE SOLUTIONS TO SDE’S WITH A JUMP NOISE 3
thistopicsee[9],Section2)thiscriteriumcanbeextendedtoanyprobabilityspacewithameasurablegroupof
admissible transformations (⇔ admissible family), that generates measurable stratification of the probability
space (for adetails, seeSection 2below). The key pointis thatthe criteriumfor the convergenceinvariations
of the family of induced measures is local in the following sense: such a convergence holds true, as soon as
the initial probability measure is restricted to any set, where the gradient (w.r.t. given admissible family) of
the limiting functional is non-degenerate. We impose a condition (condition N in Theorem 1.3 below), that
implies existenceofanadmissiblefamily,suchthatthesolutionto(0.1)possessesagradientw.r.t. this family,
and this gradient is non-degenerate with positive probability. Under this condition, the (local) criterium for
convergence in variation provides the local Doeblin condition in a small ball (Lemma 3.1 below). Together
with topological irreducibility of the process (provided, in our settings, by condition S of Theorem 1.3), this
gives condition LD.
Inourconstruction,weusetime-stretchingtransformationsoftheL´evyjumpnoise. Suchachoiceisnotthe
only possible; for instance, one can use groups of transformations, varying values of the jumps (see [8]), and
obtain another version of the non-degeneracy condition N. In order to make exposition compact, we do not
give exact formulation and proof of the correspondent statement, although the general scheme is totally the
same. Let us just outline that the main advantage of choice of the differential structure, made in the present
article, is that, for the L´evy jump noise, time-stretching transformations, unlike transformations of the phase
variable, are admissible without any regularity claim on the L´evy measure of the noise.
Thestructureofthepaperisthefollowing. InSection1,weformulatethemainstatementsofthepaper. In
Section2,wegivenecessarybackgroundfromthestochasticcalculusinvolvingtime-stretchingtransformations
oftheL´evyjumpnoise,anditsapplicationstotheconvergenceinvariationofthedistributionsofthesolutions
to SDE’s with such noise. In Section 3, we prove the main statements of the paper. Sufficient conditions for
the basic conditions R,N,S from Section 2, easy to deal with, are given in Section 4. In Section 5, we
give counterexamples showing that, in general, none of the basic conditions can be removed without losing
ergodicity of the process.
1. The main results
Letusintroducenotation. Everywherebelowν isaPoissonpointmeasureonR+×Rd,ΠisitsL´evymeasure
and ν˜(dt,du) ≡ ν(dt,du)−dtΠ(du) is the correspondent compensated point measure. We denote by p(·) the
point process associated with ν and by D the domain of p(·). Later on, we will impose such a conditions on
the coefficients a,cof the equation(0.1), that this equation, endowedby the initial condition X(0)=x∈Rm,
has the unique strong solution {X(t)≡ X(x,t),t ≥ 0}, that is a process with ca´dla´g trajectories. We denote
by P the distribution in D(R+,Rm) ofthe solution X(·)to (0.1) with Law(X(0))=µ, by E the expectation
µ µ
w.r.t. P , and by Pt the distribution of X(t) w.r.t. P . In particular,we denote P ≡P ,Pt ≡Pt ,x∈Rm.
µ µ µ x δx x δx
All the functions used below are supposed to be measurable (jointly measurable w.r.t. (x,u), if necessary).
The gradient w.r.t. the variable z is denoted by ∇z. The unit matrix in Rm is denoted by IRm. We use the
same notation k·k for the Euclidean norms both in Rm and Rd, and for an appropriate matrix norms. The
open ball in Rm with the center x and radius R is denoted by BRm(x,R). The space of probability measures
on Rm is denoted by P, the total variation norm is denoted by k·k . The (closed) support of the measure
var
µ∈P is denoted by suppµ. For µ∈P and non-negative measurable function φ:Rm →R, we denote
φ(µ)≡ φ(x)µ(dx) ∈R+∪{+∞}.
ZRm
The coefficient a is supposed to belong to the class C1(Rm,Rm) and to satisfy the linear growthcondition.
In our considerations, we will deal with the following two types of SDE’s with a jump noise.
A. Moderate non-additive noise. The SDE of the type (0.1) with the jump coefficient c dependent on
space variable x. We claim the following standard conditions to hold true:
(1.1) kc(x,u)−c(y,u)k≤K(1+kuk)kx−yk, kc(x,u)k≤ψ (x)kuk, u∈Rd,x,y ∈Rm
∗
4 ALEXEYM.KULIK
with some constant K ∈ R+ and some function ψ : Rd → R+ satisfying linear growth condition. We also
∗
claim the following specific moment condition:
(1.2) sup kc(x,u)k+k∇ c(x,u)k Π(du)<+∞, R∈R+
x Rm2
ZRdkxk≤R
(cid:2) (cid:3)
(the gradient ∇ c(x,u) is supposed to exist, and to be continuous w.r.t. x). We interpret this condition in a
x
sense that the jump part of the equation is moderate.
B. Arbitrary additive noise. TheSDEofthetype(0.1)withthejumpcoefficientcthatdoesnotdepend
on space variable x: c(x,u) =c(u) and kc(u)k≤ Kkuk. No moment conditions like (1.2) are imposed on the
jump part. In this case, (0.1) is a non-linear analogue (0.2) of the Ornstein-Uhlenbeck equation. Making a
change ν(·)7→ν (·),ν ([0,t]×A)≡ν([0,t]×c−1(A)), one can reduce (0.2) to the same equation with m =d
c c
and c(u)=u. In order to simplify notation, in a sequel we consider such an equations only.
In the both cases given above, equation (0.1), endowed by the initial condition X(0) = x, has the unique
strong solution, that is a Feller Markov process with ca´dla´g trajectories, and {Pt(·),t ∈ R+,x ∈ Rm} is its
x
transition probability. In the case A, the trajectories of this solution a.s. have bounded variation on every
finite interval.
Denote, like in [20],
Q={f ∈C2(Rm,R)|∃f˜locally bounded such that f(x+c(x,u))Π(du)≤f˜(x), x∈Rm},
Zkuk>1
and, for f ∈Q, write
Af(x)= f(x+c(x,u))−f(x)−(∇f(x),c(x,u))Rm ·1Ikuk≤1 Π(du), x∈Rm.
ZRd
(cid:2) (cid:3)
Letus formulatetwogeneralstatements concerningconvergencerateofPt to the ergodicdistributionofX
µ
and estimates for the β-mixing coefficients of X.
Theorem 1.1. Suppose that the LD condition holds true together with the following recurrence condition:
R. There exist function φ∈Q and constants α,γ >0 such that
Aφ≤−αφ+γ and φ(x)→+∞, kxk→+∞.
Then the process X possesses unique invariant distribution µ ∈ P, and there exist positive constants
inv
C ,C such that, for every µ∈P with φ(µ)<+∞,
1 2
(1.3) Pt −µ ≤C [φ(µ)+1]exp[−C t], t∈R+.
µ inv var 1 2
Recall that the β-mixing c(cid:13)oefficient f(cid:13)or X is defined by
(cid:13) (cid:13)
β (t)≡ sup E kP (·|Fs)−P (·)k , t∈R+,
µ µ µ 0 µ var,F∞
s∈R+ t+s
where Fb ≡σ(X(s),s∈[a,b]),P (·|Fs) denotes the conditional distribution of P w.r.t. Fs, and
a µ 0 µ 0
kκk ≡ sup [κ(B )−κ(B )].
var,G 1 2
B ∩B =∅,B ∪B =Rm
1 2 1 2
B ,B ∈G
1 2
If µ=µ , then β(·)≡β (·) is the mixing coefficient of the stationary version of the process X.
inv µinv
Theorem 1.2. Suppose conditions of Theorem 1.1 to hold true. Then
(i) for every µ∈P with φ(µ)<+∞
(1.4) β (t)≤C [φ(µ)+1]exp[−C t];
µ 1 2
(ii) φ(µ )<+∞, and thus the mixing coefficient β(·) allows the exponential estimate (1.4).
inv
EXPONENTIAL ERGODICITY OF THE SOLUTIONS TO SDE’S WITH A JUMP NOISE 5
In order to shorten exposition, we do not formulate here typical applications of the estimates of the type
(1.4),suchasCentralLimitTheorem,referringthe readertothe literature(see,forinstance,Theorem4[28]).
The following theorem, that gives sufficient conditions for condition LD to hold true, is the main result of
the present paper. We need some additional notation.
In the case A, put a˜(·)=a(·)− c(·,u)Π(du), and denote
kuk≤1
∆(x,uR)=[a˜(x+c(x,u))−a˜(x)]−[∇ c(x,u)]a˜(x).
x
In the case B, denote ∆(x,u)=[a(x+u))−a(x)] (note that if in the case B condition (1.2) holds true, then
these two formulas define the same function). Denote, by {Et,0 ≤ s ≤ t}, the solution to the linear SDE in
s
Rm×m
t t t
Ets =IRm + ∇a(X(r))Ersdr+ ∇xc(X(r−))Ers−ν˜(dr,du)+ ∇xc(X(r−))Ers−ν(dr,du)
Zs Zs Zkuk≤1 Zs Zkuk>1
(Et is supposed to be right continuous w.r.t. both time variables s,t), and define the random linear space S
s t
as the span (in Rm) of the set
{Et∆(X(τ−),p(τ)),τ ∈D∩(0,t)}.
τ
Theorem 1.3. Let the two following conditions to hold true.
N. There exist x ∈Rm,t >0 such that
∗ ∗
P (S =Rm)>0.
x∗ t∗
S. For any R>0 there exists t=t(R) such that
x ∈suppPt, kxk≤R.
∗ x
Then condition LD holds true.
In Section 4 below, we give some sufficient conditions for R,N,S to hold true, formulated in the terms of
the coefficients of the equation (0.1) and L´evy measure of the jump noise.
2. Time-stretching transformations and convergence in variation of induced measures
Inthis section,wegivenecessarybackgroundforthe techniqueinvolvingtime-stretchingtransformationsof
the L´evy point measure, with its applications to the problem of convergence in variation of the distributions
of the solutions to SDE’s with jumps. This technique is our tool in the proof of Theorem 1.3. Some of the
statements we give without proofs, referring to the recent papers [18],[17].
Denote H =L (R+),H =L (R+)∩L (R+),Jh(·)= ·h(s)ds,h∈H. For a fixed h∈H , we define the
2 0 ∞ 2 0 0
family {Tt,t ∈ R} of transformations of the axis R+ by putting Ttx,x ∈ R+ equal to the value at the point
h R h
s=t of the solution to the Cauchy problem
(2.1) z′ (s)=Jh(z (s)), s∈R, z (0)=x.
x,h x,h x,h
Denote T ≡ T1, then T ◦T = T ([18]). This means that T ≡ {T ,t ∈ R} is a one-dimensional
h h sh th (s+t)h h th
groupoftransformationsofthetimeaxisR+. Itfollowsfromtheconstructionthat d| T x=Jh(x),x∈R+.
dt t=0 th
We call T the time stretching transformation because, for h ∈ C(R+)∩H , it can be informally described
h 0
in the following way: every infinitesimal segment dx of the time axis should be stretched by eh(x) times, and
then all the stretched segments should be glued together, preserving initial order of the segments ([18]).
Denote Π = {Γ ∈ B(Rd),Π(Γ) < +∞} and define, for h ∈ H ,Γ ∈ Π , a transformation TΓ of the
fin 0 fin h
random measure ν by
[TΓν]([0,t]×∆)=ν([0,T t]×(∆∩Γ))+ν([0,t]×(∆\Γ)), t∈R+,∆∈Π .
h h fin
Further weusethe standardterminologyfromthe theoryofPoissonpointmeasureswithoutanyadditional
discussion. The term ”(locally finite) configuration” for a realization of the point measure is frequently used.
We suppose that the basic probability space (Ω,F,P) satisfies condition F =σ(ν), i.e. every random variable
6 ALEXEYM.KULIK
is a functional of ν. This means that in fact one can treat Ω as the configuration space over R+×(Rd\{0})
with a respective σ-algebra. The image of a configuration of the point measure ν under TΓ can be described
h
in a following way: every point (τ,x) with x 6∈ Γ remains unchanged; for every point (τ,x) with x ∈ Γ, its
“moment of the jump” τ is transformed to T τ; neither any point of the configuration is eliminated nor any
−h
new point is added to the configuration.
For h ∈ H ,Γ ∈ Π transformation TΓ is admissible for ν, i.e. the distributions of the point measures ν
0 fin h
and TΓν are equivalent ([18]). This imply that (recall that F = σ(ν)) the transformation TΓ generates the
h h
corresponding transformation of the random variables, we denote it also by TΓ.
h
Define C as the set of functionals f ∈ ∩ L (Ω,P) satisfying the following condition: for every Γ ∈ Π ,
p p fin
there exists the random element ∇Γf ∈∩ L (Ω,P,H) such that, for every h∈H ,
H p p 0
1
(2.2) (∇Γf,h) = lim [TΓ ◦f −f]
H H ε→0ε εh
with convergence in every L ,p<+∞.
p
Denote
∞
(ρΓ,h)=− h(t)ν˜(dt,Γ), h∈H ,Γ∈Π .
0 fin
Z0
Lemma 2.1. ([18], Lemma 3.2). For every Γ∈Π , the pair (∇Γ,C) satisfies the following conditions:
fin H
1) For every f ,...,f ∈C and F ∈C1(Rn),
1 n b
n
F(f ,...,f )∈C and ∇ F(f ,...,f )= F′(f ,...,f )∇ f
1 n H 1 n k 1 n H k
k=1
X
(chain rule).
2) The map ρΓ :h7→(ρΓ,h) is a weak random element in H with weak moments of all orders, and
E(∇Γf,h) =−Ef(ρΓ,h), h∈H,f ∈C
H H
(integration-by-parts formula).
3) There exists a countable set C ⊂C such that σ(C )=F.
0 0
The construction described before gives us the family T = {TΓ,h ∈ H ,Γ ∈ Π } of the admissible
h 0 fin
transformations of the probability space (Ω,F,P), such that the probabilityP is logarithmically differentiable
w.r.t. every TΓ with the correspondent logarithmic derivative equal (ρΓ,h). This allows us to introduce
h
a derivative w.r.t. such a family, that is an analogue to the Malliavin derivative on the Wiener space or
the Sobolev derivative on the finite-dimensional space. However, the structure of the family T differs from
the structure of the family of the linear shifts: for instance, there exist h,g ∈ H ,Γ ∈ Π such that
0 fin
TΓ◦TΓ 6=TΓ◦TΓ. This feature motivates the following ”refinement”, introduced in [18], of the construction
h g g h
described before.
Definition 2.2. Afamily G={[a ,b )⊂R+,h ∈H ,Γ ∈Π ,i∈N} iscalleda differential grid (orsimply
i i i 0 i fin
a grid) if
(i) for every i6=j, [a ,b )×Γ ∩ [a ,b )×Γ =∅;
i i i j j j
(ii) for every i∈N,(cid:16)Jhi >0 insi(cid:17)de ((cid:16)ai,bi) and Jh(cid:17)i =0 outside (ai,bi).
Any grid G generates a partition of some part of the phase space R+ ×(Rd\{0}) of the random measure
ν into the cells {G = [a ,b )×Γ }. We call the grid G finite, if G = ∅ for all indices i ∈ N except some
i i i i i
finite number of them. Although while studying some other problems (such as smoothness of the transition
probability density for X, see [18]) we typically use infinite grids, in our current exposition we can restrict
ourselves by a finite grids with the number of non-empty cells equal to m (recall that m is the dimension of
the phase space for the equation (0.1)). Thus, everywhere below we simplify notation from [18] and consider
the grids G with index i varying from 1 to m.
EXPONENTIAL ERGODICITY OF THE SOLUTIONS TO SDE’S WITH A JUMP NOISE 7
Denote Ti =TΓi. Forany i≤m,s,s˜∈R, the transformationsTi,Ti commute because so do the time axis
s shi s s˜
transformations T ,T . Transformation Ti does not change points of configuration outside the cell G and
shi s˜hi s i
keeps the points from this cell in it. Therefore, for every i,˜i ≤ m,s,s˜∈ R, transformations Ti,T˜i commute,
s s˜
which implies the following proposition ([18]).
Proposition 2.3. For a given grid G and t=(t ,...,t )∈Rm, define the transformation
1 m
TG =T1 ◦T2 ◦···◦Tm.
t t1 t2 tm
Then TG = {TG,l ∈ Rm} is the group of admissible transformations of Ω which is additive in the sense that
t
TG =TG◦TG,t1,2 ∈Rm.
t1+t2 t1 t2
It can be said that, by fixing the grid G, we choose from the whole family of admissible transformations
{TΓ,h ∈ H ,Γ ∈ Π } the additive sub-family, that is more convenient to deal with. The following lemma
h 0 fin
describes the differentialproperties of the solution to (0.1) w.r.t. this family. We denote by X(x,·) the strong
solution to (0.1) with X(0)=x.
Lemma 2.4. I. In the case B, for every t∈R+,Γ∈Π , every component X (t),k =1,...,m of the vector
fin k
X(t) belongs to the class C. For every h∈H ,x∈Rm, the process
0
Yh,Γ(x,t)≡((∇ΓX (x,t),h) ,...,(∇ΓX (x,t),h) )⊤, x∈Rm,t∈R+
H 1 H H m H
satisfies the equation
t t
(2.3) Yh,Γ(x,t)= ∆(X(x,s−),u)Jh(s)ν(ds,du)+ [∇a](X(x,s))Yh,Γ(x,s)ds, t≥0.
Z0 ZΓ Z0
II. In the case A, for every x∈Rm,t∈R+,Γ∈Π ,h∈H , every component of the vector X(x,t) is a.s.
fin 0
differentiable w.r.t. {TΓ,r ∈R}, i.e., there exist a.s. limits
rh
1
(2.4) Yh,Γ(x,t)= lim [TΓX (x,t)−X (x,t)], k =1,...,m.
k ε→0ε εh k k
The process Yh,Γ(x,t)=(Yh,Γ(x,t),...,Yh,Γ(x,t))⊤ satisfies the equation
1 m
Yh,Γ(x,t)= ∆(X(x,s−),u)Jh(s)ν(ds,du)+
ZZ[0,t]×Γ
t
(2.5) + [∇a](X(x,s))Yh,Γ(x,s)ds+ [∇ c](X(x,s−),u)Yh,Γ(x,s−)ν˜(ds,du), t≥0.
x
Z0 ZZ[0,t]×Rd
Statement I is proved in [18], Theorem 4.1; statement II is proved in [16], Lemma 4.1.
Remark. Solutions to equations (2.3), (2.5) can be given explicitly:
(2.6) Yh,Γ(t)= Jh(s)·Et∆(X(x,s−),u)ν(ds,du)= Jh(τ)·Et∆(X(x,τ−),p(τ)).
s τ
ZZ[0,t]×Γ τ∈DX,p(τ)∈Γ
For a given grid G = {[ai,bi) ⊂ R+,hi ∈ H0,Γi ∈ Πfin,i ≤ m}, denote YG,i = Yhi,Γi, i = 1,...,m and
consider the matrix-valued process
YG(x,t)≡(Yi(x,t))m , t∈R+.
k i,k=1
The followinglemmaisthekeypointinourapproach. Thestatementofthe lemmaisformulatedforthecases
A and B simultaneously.
Lemma 2.5. Let x∈Rm,t>0 be fixed, denote Ω ≡{detYG(x,t)6=0}. Then
x,t
(2.7) P| ◦[X(y,t)]−1−va→r P| ◦[X(x,t)]−1, y →x.
Ωx,t Ωx,t
The proof is based on the following criterium for convergence in variation of induced measures on a finite-
dimensional space, obtained in [1].
8 ALEXEYM.KULIK
Theorem 2.6. Let F,F : Rm → Rm be measurable functions, that have the approximative derivatives
n
∇F ,∇F a.s. w.r.t Lebesgue measure λm, and E ∈B(Rm) has finite Lebesgue measure. Suppose that F →F
n n
and ∇F → ∇F in a sense of convergence in measure λm and det∇F 6= 0 a.s. on E. Then the following
n
statements are equivalent.
(i) for every measurable A⊂E λm| ◦F−1−va→r λm| ◦F−1,n→+∞;
A n A
(ii) foreverymeasurableA⊂E andeveryδ >0thereexistsacompact setK ⊂Asuchthatλm(A\K )≤δ
δ δ
and lim λm(F (K ))=λm(F(K )).
n→+∞ n δ δ
In the situation, described in the preamble of the Theorem 2.6, both (i) and (ii) can fail (see, for instance,
Example1.2[17]). Thus,inordertoprovide(i)(thatisourgoal),weshouldimposesomeadditionalconditions
on the sequence {F }, sufficient for (ii) to hold true. The following two sufficient conditions were proved in
n
[1], Corollaries 2.5 and 2.7, and in [17], Theorem 3.1, correspondingly.
Proposition 2.7. I. Let FF ∈ W1 (Rm,Rm) with p ≥ m (W1 denotes the local Sobolev space), and
, n p,loc p,loc
Fn →F,n→∞ w.r.t. Sobolev norm k·kW1(Rm,Rm) on every ball. Then
p
(2.8) λm| ◦F−1−va→r λm| ◦F−1,n→+∞ for every measurable A⊂{det∇F 6=0}.
A n A
II. Let in the situation, described in the preamble of the Theorem 2.6, the sequence {F } be uniformly
n
approximatively Lipschitz. This, by definition, means that for every δ >0,R<+∞ there exist a compact
set Kδ,R and a constant Lδ,R <+∞ such that λm(BRm(0,R)\Kδ)<δ and every function Fn|Kδ is a Lipschitz
function with the Lipschitz constant L . Then (2.8) holds true.
δ,R
Proof of the Lemma 2.5. Take the sequence y → x,n → +∞, and denote f = X(x,t),f = X(y ,t).
n n n
It is proved in [18] (proof of Theorem 3.1), that the group TG ≡ {TG,s ∈ Rm} generates a measurable
s
parametrization of (Ω,F,P), i.e. there exists a measurable map Φ : Ω → Rm ×Ω˜ such that Ω˜ is a Borel
measurable space and the image of every orbit of the group TG under Φ has the form L×{̟}, where ̟ ∈Ω˜
and L is a linear subspace of Rm. The linear subspace L differs from Rm exactly in the case, when the orbit
TG{ω} is built for such an ω, that, for some i = 1,...,m, ν((a ,b )×Γ ) = 0 (i.e., some Ti does not change
i i i
ω). For every ω of such a type detYG(x,t) = 0, and thus we need to investigate the laws of f,f , restricted
n
to ΩG ≡{ν((ai,bi)×Γi)>0,i=1,...,m}, only.
The measure P| can be decomposed into a regular family of conditional distributions such that every
ΩG
conditional distribution is supported by an orbit of the group TG (see, for instance, [23]). Therefore we can
write
(2.9) P(A)= P̟([A]̟)π(̟), A∈F∩ΩG,
ZΩ˜
where [A] = {s ∈ Rm|(s,̟) ∈ A}, π is the image of P| under the natural projection Ω → Ω˜ and P(·)
̟ ΩG ·
is a probabilistic kernel, i.e. P(A) is a measurable function for every A ∈ B(Rm) and P (·) is a probability
· ̟
measure on Rm for every ̟ ∈Ω˜. Any functional g on ΩG now can be considered as a functional on Rm×Ω˜,
g = {g(s,̟),s ∈ Rm,̟ ∈ Ω˜}. Below, we denote [g] (·) ≡ g(·,̟) : Rm → Rm, ̟ ∈ Ω˜. One can write, for
̟
A⊂ΩG, that
(P| ◦g−1)(·)= [P | ◦[g]−1](·)π(d̟).
A ̟ A̟ ̟
ZΩ˜
Therefore, in order to prove the statement of Lemma 2.5, it is sufficient to prove that
(2.10) P | ◦[f ] −va→r P | ◦[f] for π-almost all ̟ ∈P˜.
̟ [Ωx,t]̟ n ̟ ̟ [Ωx,t]̟ ̟
Denote ρi =(ρΓi,hi),i=1,...,m, andlet ρG be Rm-valuedfunction such that (ρG,t)= mi=1tiρi,t∈Rm.
Thenonecanshowthat,forπ-almostall̟ ∈Ω˜,the measureP possessesthe logarithmicderivativeequalto
̟ P
[ρG] (wedonotgivethedetailedexpositionhere,sincethisfactisquiteanalogoustotheoneforlogarithmically
̟
EXPONENTIAL ERGODICITY OF THE SOLUTIONS TO SDE’S WITH A JUMP NOISE 9
differentiable measures on linear spaces, see [5]). One can deduce from the explicit formula for ρG that there
exists c>0 such that Eexp[(ρG,t)]<+∞,ktk≤c, and, therefore, that for π-almost all ̟ ∈Ω˜
(2.11) exp[([ρG] (s),t)]P (ds)<+∞, ktk≤c.
̟ ̟
ZRm
Due to Proposition4.3.1 [4], for every ̟ such that (2.11) holds true, the measure P has the form P (dx)=
̟ ̟
p (x)λm(dx), where the function p is continuous and positive. Therefore, in order to prove (2.10), it is
̟ ̟
enough to prove that, for π-almost all such ̟ and every R>0,
(2.12) λm| ◦[f ] −va→r λm| ◦[f] for π-almost all ̟ ∈P˜.
BRm(0,R)∩[Ωx,t]̟ n ̟ BRm(0,R)∩[Ωx,t]̟ ̟
In order to prove (2.12), let us introduce auxiliary notions and give their relations with the notions of
Sobolev and approximative derivatives.
Definition 2.8. For a given function F : Rm → Rm and σ-finite measure κ on B(Rm) we say that F is
direction-wise κ-a.s. differentiable, if there exists function ∇F : Rm → Rm×m such that, for every t ∈ Rm,
1[F(·+tε)−F(·)] → (∇F(·),t),ε → 0 κ-a.s. We say that g is direction-wise differentiable in the L (κ)
ε p,loc
sense, if F ∈L (κ) and, for every t∈Rm, 1[F(·+tε)−F(·)]→(∇F(·),t),ε→0 in L (κ).
p,loc ε p,loc
Proposition 2.9. 1. Let κ(dx) = p(x)λm(dx) with p(x) ≥ C > 0,kxk ≤ R for any R > 0. Then
R
every function F, that is direction-wise κ-a.s. differentiable, is also direction-wise λm-a.s. differentiable, and
every function F, that is direction-wise differentiable in L (κ) sense, is also direction-wise differentiable in
p,loc
L (λm) sense. The function ∇F from the definition of κ-differentiability (either in a.s. or L sense)
p,loc p,loc
λm-a.s. coincides with the one from the definition of λm-differentiability.
2. If F is direction-wise L (λm)-differentiable, then F ∈ W (Rm,Rm) and ∇F coincides with its
p,loc p,loc
Sobolev derivative.
3. If F is direction-wise λm-a.s. differentiable, then F has approximative derivative at λm-almost all points
x∈Rm, and ∇F coincides with its approximative derivative.
Proof. Statements 1 and 2 immediately follow from the definition. Statement 3 follow from Theorem
3.1.4 [11] and the trivial fact, that the usual differentiability at some point w.r.t. given direction implies
approximative differentiability at the same point w.r.t. this direction.
Now, we can finish the proof of Lemma 2.5. Denote f = [f ] ,f = [f] . By the construction,
n,̟ n ̟ ̟ ̟
[TGf ] (s) = [f ] (s +r), s,r ∈ Rm,̟ ∈ Ω˜. This, together with (2.9) and Lemma 2.4, provides that
r n ̟ n ̟
there exists Ω˜ ⊂ Ω˜ with π(Ω˜\Ω˜ ) = 0 such that, for every ̟ ∈ Ω˜ , (2.11) holds true, and the functions
0 0 0
f ,f are either direction-wise P -differentiable (in the case A), or direction-wise differentiable in the
n,̟ ̟ ̟
L (P ) sense (in the case B), and ∇f = [YG(y ,t)] ,∇f = [YG(x,t)] . This, in particular, means
p,loc ̟ n,̟ n ̟ ̟ ̟
that [Ω ] ={det∇f 6=0} for ̟ ∈Ω˜ .
x,t ̟ ̟ 0
Now,wecanapplythe standardtheoremonL -continuityofthesolutiontoanSDEw.r.t. initialcondition
p
(see Theorem 4, Chapter 4.2 [12]), and obtain that f → f,[YG(y ,t)] → [YG(x,t)],n → ∞, in L sense.
n n p
This implies that, for π-almost all ̟ ∈ Ω˜ , f → f ,∇f → ∇f ,n → ∞ in L (P ) sense, and,
0 n,̟ ̟ n,̟ ̟ p ̟
therefore, in L (λm) sense. In the case B, convergence (2.12) (and thus the statement of the lemma)
p,loc
follows straightforwardly from the statement I of Proposition 2.7. In the case A, convergence (2.12) follows
fromthestatementIIofthesameProposition,andLemma3.3[17],thatprovidesthat,forπ-almostall̟ ∈Ω˜,
the sequence {∇f } is uniformly approximatively Lipschitz on Rm. The lemma is proved.
n,̟
3. Proofs of the main results
3.1. Proof of Theorem 1.3. We prove Theorem 1.3 in two steps. First, we use Lemma 2.5 and show that,
under condition N, the local Doeblin condition holds true inside some small ball.
Lemma 3.1. Under condition N, there exists ε >0, such that
∗
10 ALEXEYM.KULIK
(3.1) inf Pt∗ ∧Pt∗ (dz)>0.
x y
x,y∈B(x∗,ε∗)ZRm
(cid:2) (cid:3)
Proof. Suppose that the grid G is such that, in the notation of Lemma 2.5,
(3.2) P(Ω )>0,
x∗,t∗
and denote P (dz)=P (Ω ,X(x,t )∈dz). One can see that P (dz)=p (z)Pt∗(dz) with p ≤
x∗,x x x∗,t∗ ∗ x∗,x x∗,x x x∗,x
1, and therefore
[P ∧P ](dz)≤ Pt∗ ∧Pt∗ (dz), x,y ∈Rm.
x∗,x x∗,y x y
ZRm ZRm
Thus, in order to prove (3.1), it is enough to prove that(cid:2) (cid:3)
inf [P ∧P ](dz)>0.
x∗,x x∗,y
x,y∈B(x∗,ε∗)ZRm
The latter inequality follows from the condition (3.2), Lemma 2.5 and relation
1
[P ∧P ](dz)=P(Ω )− kP (·)−P (·)k →P(Ω ), x,y →x .
ZRm x∗,x x∗,y x∗,t∗ 2 x∗,x x∗,y var x∗,t∗ ∗
Thus, the onlything leftto showis that, underconditionN,the gridGcanbe choseninsuchawaythat(3.2)
holds true. Denote, by J , the family of all rational partitions of (0,t ) of the length 2m; any J ∈ J is the
m ∗ m
set of the type
J ={a ,b ,a ,b ,...,a ,b }, with 0<a <b <···<a <b <t , a ,b ∈Q, i=1,...,m.
1 1 2 2 m m 1 1 m m ∗ i i
Denote, for the set J of such a type and r ∈N,
1
Ω = ∀i=1,...,m ∃!τ ∈(a ,b )∩D, kp(τ )k≥ , i=1,...,m
J,r i i i i
r
(cid:26)
and span{Et ∆(X(x ,τ −),p(τ )),i=1,...,m}=Rm .
τi ∗ i i
(cid:27)
Then, elementary considerations show that
Ω = Ω .
x∗,t∗ J
J∈J[m,r∈N
Therefore, under condition N, there exist J∗ ={a∗1,b∗1,...,a∗m,b∗m} and r∗ ∈N such that P(ΩJ∗,r∗)>0. Let
h∗ ∈H ,i=1,...,m be arbitrary functions such that, for any i, Jh∗ >0 inside (a∗,b∗) and Jh∗ =0 outside
i 0 i i i i
(a∗,b∗). Consider the grid G∗ with
i i
{u|kuk≥ 1}, i≤m
Γ = r , a =a∗,b =b∗,h =h∗, i≤m, a ,b ,h are arbitrary for i>m.
i i i i i i i i i i
(∅, i>m
Then formula (2.6) shows that, on the set ΩJ∗,r∗,
YG∗,i(x ,t )≡Yhi,Γi(x ,t )=Jh (τ )E ∆(X(x ,τ −),p(τ )), i=1,...,m.
∗ ∗ ∗ ∗ i i τi ∗ i i
SinceJh (τ )6=0bytheconstruction,thevectors{YG∗,i(x ,t )}arelinearlyindependentiffsoarethevectors
i i ∗ ∗
{E ∆(X(x ,τ −),p(τ ))}. Thus,
τi ∗ i i
P(Ωx∗,t∗)=P(detY(x∗,t∗)6=0)≥P(ΩJ∗,r∗)>0,
that gives (3.2). The lemma is proved.
ThelaststepintheproofofTheorem1.3istocombinestatementofthepreviouslemmawiththecondition
S and show, that the local Doeblin condition holds true in any bounded region of Rm.
Lemma 3.2. Under conditions N and S, condition LD holds true with
T(R)=t(R)+t , R>0.
∗